Quantum hardware characterized by programmable bose-hubbard hamiltonians

ABSTRACT

An apparatus includes a first group of superconducting cavities and a second group of superconducting cavities, each of which is configured to receive multiple photons. The apparatus includes couplers, where each coupler couples one superconducting cavity from the first group with one cavity from the second group such that the photons in the coupled superconducting cavities interact. A first superconducting cavity of the first group is connected to a second superconducting cavity of the second group, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities. The first superconducting cavity is coupled to at least one other superconducting cavity of the first group to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to at least one other superconducting cavity of the second group to which the first superconducting cavities are coupled.

BACKGROUND

The present specification relates to quantum hardware characterized byprogrammable Bose-Hubbard Hamiltonians.

SUMMARY

In a computational paradigm of this specification, quantum informationis represented by multimode quantum hardware, the dynamics of which canbe characterized and controlled by a programmable many-body quantumHamiltonian. The multimode quantum hardware can be programmed as, forexample, a quantum processor for certain machine learning problems.Examples of the quantum hardware include neutral atoms on opticallattices, photonic integrated circuits, or superconducting cavityquantum electrodynamics (QED) circuits, and the Hamiltonianscharacterizing such quantum hardware include dissipative ornon-dissipative Bose-Hubbard Hamiltonians.

The solution to a machine optimization problem can be encoded into anenergy spectrum of a Bose-Hubbard quantum Hamiltonian. For example, thesolution is encoded in the ground state of the Hamiltonian. Through anannealing process in which the Hamiltonian evolves from an initialHamiltonian into a problem Hamiltonian, the energy spectrum or theground state of the Hamiltonian for solving the problem can be obtainedwithout diagonalizing the Hamiltonian. The annealing process may notrequire tensor product structure of conventional qubits or rotations andmeasurements of conventional local single qubits. In addition, quantumnoise or dechoerence can act as a recourse to drive the non-equilibriumquantum dynamics into a non-trivial steady state. The quantum hardwarecan be used to solve a richer set of problems as compared to quantumhardware represented by an Ising Hamiltonian. Furthermore, instead ofthe binary representations provided by the Ising Hamiltonians,constraint functions of problems to be solved can have a digitalrepresentation according to the density of states in Cavity QED modes.

In general, in some aspects, the subject matter of the presentdisclosure can be embodied in apparatuses that include: a first group ofsuperconducting cavities each configured to receive multiple photons; asecond group of superconducting cavities each configured to receivemultiple photons; and multiple couplers, in which each coupler couplesone superconducting cavity from the first group of superconductingcavities with one superconducting cavity from the second group ofsuperconducting cavities such that the photons in the coupledsuperconducting cavities interact, and in which a first superconductingcavity of the first group of superconducting cavities is connected to asecond superconducting cavity of the second group of superconductingcavities, such that photons of the first and second superconductingcavities are shared by each of the first and second superconductingcavities, the first superconducting cavity is coupled to one or more ofthe other superconducting cavities of the first group of superconductingcavities to which the second superconducting cavities are coupled, andthe second superconducting cavity is coupled to one or more of the othersuperconducting cavities of the second group of superconducting cavitiesto which the first superconducting cavities are coupled.

Various implementations of the apparatuses are possible. For example, insome implementations, each coupler is configured to annihilate a photonin one superconducting cavity and create a photon in a differentsuperconducting cavity.

In some implementations, at least one of the couplers includes aJosephson junction.

In some implementations, a Hamiltonian characterizing the apparatus is:Σ_(i)h_(i)n_(i)+Σ_(i,j)t_(ij)(α_(i)^(†)α_(j)+h.c.)+Σ_(i)U_(i)n_(i)(n_(i)−1), in which n_(i) is a particlenumber operator and denotes occupation number of a cavity mode i, α_(i)^(†) is a creation operator that creates a photon in cavity mode i,α_(i) is an annihilation operator that annihilates a photon in cavitymode j, h_(i) corresponds to a site disorder, U_(i) corresponds to anon-site interaction, t_(i,j) are the hopping matrix elements, and h.c.is hermitian conjugate. In some implementations, the multiple couplersare trained to produce an output desired probability density function ata subsystem of interest at an equilibrium state of the apparatus. Insome implementations, the apparatuses are trained as Quantum Boltzmannmachines.

In some implementations, wherein a Hamiltonian characterizing theapparatus is: Σ_(i)h_(i)n_(i)+Σ_(i,j)t_(ij)(α_(i)^(†)α_(j)+h.c.)+Σ_(i)U_(i)n_(i)(n_(i)−1)+Σ_(i,j)U_(ij)n_(i)n_(j), inwhich n_(i) is a particle number operator and denotes occupation numberof a cavity mode i, α_(i) ^(†) is a creation operator that creates aphoton in cavity mode i, α_(i) is an annihilation operator thatannihilates a photon in cavity mode j, h_(i) corresponds to a sitedisorder, U_(i) corresponds to an on-site interaction, t_(i,j) are thehopping matrix elements, and h.c. is hermitian conjugate. Theapparatuses can be operable to evolve adiabatically to a ground state ofa problem Hamiltonian H_(p)=Σ_(i)h_(i)n_(i)+Σ_(i)U_(i)n_(i)(n_(i)+1)+Σ_(i,j)U_(ij)n_(i)n_(j). Theapparatus can be operable to evolve adiabatically from a Mott-insulatorstate to a superfluid state, in which an initial Hamiltonian of theapparatus is H_(i)=Σ_(i,j)t_(ij)(α_(i) ^(†)α_(j)+h.c.). The apparatuscan be operable to evolve adiabatically from a Mott-insulator state to aground state of a problem HamiltonianH_(p)=Σ_(i)h_(i)n_(i)+Σ_(i)U_(i)n_(i)(n_(i)−1)+Σ_(i,j)U_(ij)n_(i)n_(j),in which an initial Hamiltonian of the apparatus isH_(i)=Σ_(i,j)t_(ij)(α_(i) ^(†)+h.c.).

In some implementations, the apparatus is configured to respond to anexternal field ε(t) and a Hamiltonian characterizing the apparatus inthe external field is: Σ_(i)h_(i)n_(i)+Σ_(i,j)t_(ij)(α_(i)^(†)α_(j)+h.c.)+Σ_(i)U_(i)n_(i)(n_(i)−1)+Σ_(i)[ε(t)α_(i)^(†)+ε(t)*α_(i)]+H_(SB), in which H_(SB)=Σ_(i)Σ_(υ)[κ_(i,υ)(α_(i)b_(υ)^(†)+α_(i) ^(†)b_(υ))]+λ_(i,υ)α_(i) ^(†)α_(i)(b_(υ)+b_(υ) ^(†)), and inwhich n_(i) is a particle number operator, ε(t) is a slowly-varyingenvelope of an externally applied field to compensate for photon loss,H_(SB) is a Hamiltonian of the interaction between the apparatus and abackground bath in which the apparatus is located, b_(υ) and b_(υ) ^(†)are annihilation and creation operators for a bosonic background bathenvironment, κ_(i,υ) is a strength of apparatus-bath interactionscorresponding to exchange of energy, h_(i) corresponds to a sitedisorder, U_(i) corresponds to an on-site interaction, t_(i,j) are thehopping matrix elements, and λ_(i,υ) corresponds to a strength of localphoton occupation fluctuations due to exchange of phase with the bath.The apparatus can be operable to be dissipatively-driven to a groundstate of a problem Hamiltonian.

In some implementations, at least one cavity is a 2D cavity. Forexample, each cavity can be a 2D cavity.

In some implementations, at least one cavity is a 3D cavity. Forexample, each cavity can be a 3D cavity.

In some implementations, each superconducting cavity in the first groupof superconducting cavities is connected to a superconducting cavity inthe second group of superconducting cavities.

In general, in other aspects, the subject matter of the presentdisclosure can be embodied in methods that include providing anapparatus having: a first group of superconducting cavities eachconfigured to receive multiple photons; a second group ofsuperconducting cavities each configured to receive multiple photons;and multiple couplers, in which each coupler couples one superconductingcavity from the first group of superconducting cavities with onesuperconducting cavity from the second group of superconducting cavitiessuch that the photons in the coupled superconducting cavities interact,and in which a first superconducting cavity of the first group ofsuperconducting cavities is connected to a second superconducting cavityof the second group of superconducting cavities, such that photons ofthe first and second superconducting cavities are shared by each of thefirst and second superconducting cavities, the first superconductingcavity is coupled to one or more of the other superconducting cavitiesof the first group of superconducting cavities to which the secondsuperconducting cavities are coupled, and the second superconductingcavity is coupled to one or more of the other superconducting cavitiesof the second group of superconducting cavities to which the firstsuperconducting cavities are coupled. The apparatus can be provided inan initial Mott-insulated state. The methods can further include causinga quantum phase transition of the apparatus from the initialMott-insulator state to a superfluid sate; and adiabatically guiding theapparatus to a problem Hamiltonian.

Various implementations of the methods are possible. For example, insome implementations, the methods can further include causing a quantumphase transition of the apparatus from the superfluid state to a finalMott-insulator state and reading the state of each superconductingcavity in the apparatus. The details of one or more embodiments of thesubject matter of this specification are set forth in the accompanyingdrawings and the description below. Other features, aspects, andadvantages of the subject matter will become apparent from thedescription, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of an example structure of quantum hardware.

FIG. 2A is an example of a selected connection in quantum hardware.

FIG. 2B is an example of a full connection in quantum hardware.

FIG. 3 is a flow diagram of an example process for encoding a problem ina Hamiltonian containing density-density interactions and programmingquantum hardware.

FIG. 4 is a flow diagram of an example process for encoding a problem ina dissipative-driven Hamiltonian and programming quantum hardware.

DETAILED DESCRIPTION

FIG. 1 is a schematic of an example structure of quantum hardware 100that can be characterized by programmable Bose-Hubbard Hamiltonians. Thequantum hardware 100 includes QED cavities 104 arranged in columns 110,and lines 112. At least some pairs of the QED cavities, such as cavities110 and 112, are coupled to each other through coupler 106. The QEDcavities can be superconducting waveguide cavities restricted indimensionality, e.g., to 1D, 2D or 3D. The couplers 106 can be inductivecouplers, and the hardware can be configured with resistors andinductors. The couplers 106 can be Josephson couplers, and in anexample, a Josephson coupler is constructed by connecting twosuperconducting elements separated by an insulator and a capacitance inparallel.

The cavities contain photons in optical modes 102. The cavities canreceive a variable amount of photons when the quantum hardware isinitialized, or during the use of the quantum hardware. A coupler 106between two cavities allows the photons of the two cavities to interactwith each other. For example, the coupler can create or annihilatephotons in a cavity, or move photons between cavities. Each cavity inthe hardware 100 can be used as a logical computation unit. The numberof photons in a cavity mode of the cavity can be read using photondetectors 108.

In some implementations, the quantum hardware 100 includes a fullyconnected network of superconducting cavities 104. In this network, eachcavity is coupled with all other cavities through couplers 106. In otherimplementations, selected pairs of cavities are coupled with each other.The selection can be made based on the need for the quantum computationand the physical confinement of the hardware.

FIG. 2A is an example of a selected connection in quantum hardware. Thehardware includes cavities A, B, C, D, E, F, G, and H, and pairs ofcavities are coupled through couplers 208. Cavities “A” 202 and “E” 204are selected to be connected through a connection 206, so thateffectively, they become the same cavity. That is, the connection can beconsidered an extension of the cavity QED mode. Without the connection206, the cavity “A” is coupled to cavities “E”, “F”, “G”, and “H,” butnot to cavities B, C, and D.

Effectively, in this example, cavity “A” is coupled to all othercavities of the hardware. However, cavities “B”-“H” are only coupled toselected cavities of the hardware. To increase the number of cavitieseach cavity is coupled to, additional connections similar to theconnection 206 can be added. The total amount of interaction betweencavities in the hardware can be increased.

An example of a fully connected network is shown in FIG. 2B. FIG. 2Bincludes connections between “E” and “A” 212, “F” and “B” 214, “G” and“C” 216, and “H” and “D” 218. The network of FIG. 2B therefore allowsfor each cavity to interact with all other cavities.

The hardware of FIGS. 1, 2A, and 2B can be characterized by aBose-Hubbard Hamiltonian:

$H = {{\sum\limits_{i}{h_{i}n_{i}}} + {\sum\limits_{i,j}{t_{ij}\left( {{a_{i}^{\dagger}a_{j}} + {h.c.}} \right)}} + {\sum\limits_{i}{U_{i}{n_{i}\left( {n_{i} - 1} \right)}}}}$

where n_(i) is the particle number operator and denotes the occupationnumber of a cavity mode i, α_(i) ^(†) is a creation operator thatcreates a photon in cavity mode i, α_(j) is an annihilation operatorthat annihilates a photon in cavity mode j, h_(i) corresponds to a sitedisorder, U_(i) corresponds to an on-site interaction, are the hoppingmatrix elements, and h.c. is the hermitian conjugate.

The hardware of FIGS. 1, 2A, and 2B, characterized by the Bose-HubbardHamiltonian above, can be used to determine solutions to problems bytraining the hardware as a Quantum Boltzmann Machine for probabilisticinference on Markov Random Fields. For example, a problem can be definedby a set of observables y_(i), e.g., photon occupation number at acavity of the hardware, and a goal is to infer underlying correlationsamong a set of hidden variables x_(i). Assuming statistical independenceamong various pairs of y_(i) and x_(i), the joint probabilitydistribution would be

${{p\left( {\left\{ x_{i} \right\},\left\{ y_{i} \right\}} \right)} = {{1/Z}{\prod\limits_{i,j}\; {{\theta_{i,j}\left( {x_{i},x_{j}} \right)}{\prod\limits_{i}\; {\alpha_{i}\left( {x_{i},y_{i}} \right)}}}}}},$

where Z is the partition function (which, for a given system with afixed energy function or a given Hamiltonian, is constant),θ_(i,j)(x_(i), x_(j)) is a pairwise correlation, and α_(i)(x_(i), y_(i))is the statistical dependency between a given pair of y_(i) and x_(i).

In training the hardware, certain cavity modes can act as thevisible/observable input nodes of the Markov Random Field and can beused to train one or more of the Josephson couplers, which connect thehidden nodes x_(i), to reproduce certain probability distribution ofoutcomes at the output visible nodes y_(i).

For example, the training can be such that the delocalized energy groundstate of the Bose-Hubbard model for each input state can have aprobability distribution over the computational, i.e., localized, basisthat resembles the output probability distribution function (PDF) of thetraining example, i.e., p({x_(j)}, {y_(i)}). Thus, the thermalized stateof the hardware trained as a Quantum Boltzmann Machine can be sampled toprovide a probabilistic inference on the test data according to theBoltzmann distribution function.

For example, an energy function can be defined:

${E\left( {\left\{ x_{i} \right\},\left\{ y_{i} \right\}} \right)} = {{- {\sum\limits_{i,j}\; {t_{i,j}\left( {x_{i},x_{j}} \right)}}} - {\sum\limits_{i}\; {h_{i}\left( {x_{i},y_{i)}} \right.}}}$

where the nonlocal pairwise interactions t_(i,j)(x_(i),x_(j))=θ_(i,j)(x_(i), x_(j)), and disordered local fields h_(i)(x_(i),y_(i))=α_(i)(x_(i), y_(i)).

The Boltzmann distribution of the above energy function is then:

${{p\left( {\left\{ x_{i} \right\},\left\{ y_{i} \right\}} \right)} = {\frac{1}{Z}^{- \frac{E{({{\{ x_{i}\}},{\{ y_{i}\}}})}}{T}}}},$

where Z is the partition function.

In some other implementations, the quantum hardware of FIGS. 1, 2A, and2B can be engineered or controlled to allow an additional type ofcoupling between the coupled cavities characterized by density-densityinteractions. With density-density interactions, an additional term canbe added to the Bose-Hubbard Hamiltonian:

${\sum\limits_{i,j}\; {U_{ij}n_{i}n_{j}}},$

The addition of the density-density interaction term to the Bose-HubbardHamiltonian can allow construction of a problem Hamiltonian in which thesolution of a wide variety of problems can be encoded. For example,constraint functions of problems can have a digital representationaccording to the density of states in cavity QED modes withdensity-density interactions.

To account for photon loss in the above Hamiltonian, in someimplementations the hardware is driven with additional fields tocompensate for the loss.

To provide the density-density interaction term in the Hamiltonian, thequantum hardware can be engineered (an example process of using theadditionally engineered hardware is shown in FIG. 3) or by controllingthe hardware using a dissipative-driven method (an example process ofusing the controlled hardware is shown in FIG. 4).

Using the quantum hardware of FIG. 1 as an example, the quantum hardwarecan additionally be engineered to include, e.g., Kerr non-linearity withJosephson Junction couplers, the Stark effect, or continuous-timeC-phase gates between cavities.

The modified Hamiltonian characterizing the additionally engineeredhardware is therefore:

${H_{total} = {{\sum\limits_{i}\; {h_{i}n_{i}}} + {\sum\limits_{i,j}\; {t_{ij}\left( {{a_{i}^{\dagger}a_{j}} + {h.c.}} \right)}} + {\sum\limits_{i}\; {U_{i}{n_{i}\left( {n_{i} - 1} \right)}}} + {\sum\limits_{i,j}\; {U_{ij}n_{i}n_{j}}}}},$

where the final term is the density-density interactions betweencavities i and j.

In use for adiabatic computation, a time dependent Hamiltonian can berepresented as:

H _(total)=(1−s)H _(i) +sH _(p),

where s is a control parameter and can be a linear function of time,H_(i) is the initial Hamiltonian:

${H_{i} = {\sum\limits_{i,j}\; {t_{ij}\left( {{a_{i}^{\dagger}a_{j}} + {h.c.}} \right)}}},$

and H_(p) is the problem Hamiltonian into which the selected problem isencoded:

H _(p)=Σ_(i) h _(i) n _(i)+Σ_(i) U _(i) n _(i)(n _(i)−1)+Σ_(i,j) U _(ij)n _(i) n _(j).

When s=0, the hardware is placed into an initial ground state that isknown. The hardware is then quais-adiabatically guided to s=1, movingthe hardware to the ground state of the Hamiltonian encoded by theproblem.

FIG. 3 is a flow diagram of an example process 300 for encoding aproblem in a Hamiltonian containing density-density interactions andprogramming quantum hardware.

In solving a given problem, e.g., an optimization problem, a problemmodeled as a Markov Random Field, or an NP-Hard problem, the hardwareundergoes a quantum phase transition from a Mott-insulator state to asuperfluid state (step 302). The hardware is initially in an insulatedstate with no phase coherence, and with localized wavefunctions only.The many-body state is therefore a product of local Fock states for eachcavity in the hardware:

|ψ_(MI)

=Π_(i)α_(i) ^(N)|0

,

where N is the number of photons, and i is the cavity mode.

The hardware undergoes a quantum phase transition to a superfluid stateso that the wavefunctions are spread out over the entire hardware:

${\psi_{SF}\rangle} = {\sum\limits_{i}\; {a_{i}^{N}{0\rangle}}}$

The hardware is adiabatically guided to a problem Hamiltonian (step304). That is, the hardware is moved from the s=0 state, to the s=1state as explained above.

At the end of the annealing process, the hardware transitions from asuperfluid state to a non-trivial Mott-insulator state that can capturethe solution to the problem (step 306).

The quantum state of the entire hardware is read out (step 308) and canbe processed by a classical computer to provide solutions to the givenproblem. For example, the state of each cavity is determined by thephoton occupation number of each cavity mode. The process 300 can berepeated multiple times for the given problem to provide solutions witha statistical distribution.

Alternatively, using the quantum hardware of FIG. 1 as an example, thedynamical effects of density-density interactions can be achieved by aninterplay of the Bose-Hubbard Hamiltonian with cavity photon numberfluctuations induced by an auxiliary external field. The combination ofthe hardware and the auxiliary external field is called adissipative-driven hardware, and the Hamiltonian describing thedissipative-driven hardware is:

$H_{BH} = {{\sum\limits_{i}\; {h_{i}n_{i}}} + {\sum\limits_{i,j}\; {t_{ij}\left( {{a_{i}^{\dagger}a_{j}} + {h.c.}} \right)}} + {\sum\limits_{i}\; {U_{i}{n_{i}\left( {n_{i} - 1} \right)}}} + {\sum\limits_{i}\; \left\lbrack {{{ɛ(t)}a_{i}^{\dagger}} + {{ɛ(t)}^{*}a_{i}}} \right\rbrack} + H_{SB}}$

where ε(t) is a slowly-varying envelope of an externally applied fieldto compensate for photon loss, and H_(SB) is the Hamiltonian of theinteraction between the hardware and the background bath in which thehardware is located:

${H_{SB} = {\sum\limits_{i}{\sum\limits_{v}\left\lbrack {{\kappa_{i,v}\left( {{a_{i}b_{v}^{\dagger}} + {a_{i}^{\dagger}b_{v}}} \right)} + {\lambda_{i,v}a_{i}^{\dagger}{a_{i}\left( {b_{v} + b_{v}^{\dagger}} \right)}}} \right\rbrack}}},$

where b_(υ), and b_(υ) ^(†) are annihilation and creation operators forthe bosonic background bath environment, κ_(i,υ) is the strength ofhardware-bath interactions corresponding to the exchange of energy,h_(i) corresponds to a site disorder, U_(i) corresponds to an on-siteinteraction, t_(i,j) are the hopping matrix elements, and λ_(i,υ)corresponds to the strength of local photon occupation fluctuations dueto exchange of phase with the bath.

Using the dissipative-driven hardware, a solution to a problem can bedetermined without adiabatically guiding the hardware to the groundstate of a problem Hamiltonian as in the process 300 of FIG. 3. Thedissipative-driven hardware is eventually dominated by dissipativedynamics, defining a non-trivial steady state in which the solution to aproblem is encoded.

FIG. 4 is a flow diagram of an example process 400 for encoding aproblem in a dissipative-driven Hamiltonian and programming quantumhardware.

The hardware is programmed for a problem to be solved (step 402). Insome implementations the problem is an optimization problem or aninference task and is mapped to a Markov Random Field. For example, aproblem can be defined by a set of observables y_(i), e.g., photonoccupation number at a cavity of the hardware, and the goal is to inferunderlying correlations among a set of hidden variables x_(i). Assumingstatistical independence among pair y_(i) and x_(i), the jointprobability distribution would be:

${{p\left( {\left\{ x_{i} \right\},\left\{ y_{i} \right\}} \right)} = {{1/Z}{\prod\limits_{i,j}\; {{\theta_{i,j}\left( {x_{i},x_{j}} \right)}{\prod\limits_{i}\; {\alpha_{i}\left( {x_{i},y_{i}} \right)}}}}}},$

where Z is a normalization constant, θ_(i,j)(x_(i), x_(j)) is a pairwisecorrelation, and α_(i)(x_(i), y_(i)) is the statistical dependencybetween a given pair of y_(i) and x_(i).

In many classes of machine learning problems, e.g., computer vision,image processing, and medical diagnosis, the goal of the problems is tocompute marginal probabilities:

${p\left( x_{N} \right)} = {\sum\limits_{x\; 1}\; {\sum\limits_{x\; 2}\; {\ldots \mspace{14mu} {\sum\limits_{x_{N - 1}}\; {p\left( {\left\{ x_{i} \right\},\left\{ y_{i} \right\}} \right)}}}}}$

Using a density-matrix formulation, the marginal probabilities can becomputed from the dynamics of the dissipative-driven hardware in aquantum trajectory picture:

${\frac{\rho}{t} = {{- {i\left\lbrack {{H_{BH} + H_{LS} + H_{decoh}},\rho} \right\rbrack}} + {\sum\limits_{i,i^{\prime},j,j^{\prime}}\; {\Gamma_{i,i^{\prime},j,j^{\prime}}a_{i}^{\dagger}a_{i^{\prime}}\rho \; a_{j^{\prime}}^{\dagger}a_{j}}} + {\sum\limits_{i}\; {\Lambda_{i}a_{i}\rho \; a_{i}^{\dagger}}}}},$

where [ ] is the commutator, ρ is the density matrix, H_(BH) is theHamiltonian describing the dissipative-driven hardware, H_(LS) is theLamb shift, H_(decoh) is an anti-Hermitian term proportional to thedechoerence rate of the hardware that leads to relaxation in the fixedexcitation manifold and can be the Fourier transform of the bathcorrelation functions; Γ_(i,i′,j,j′), is a tensor describing the quantumjump rate among fixed-excitation manifolds, and Λ_(i) is a tensordescribing quantum jump rates between fixed-excitation manifolds.

The dechoerence of the hardware is gradually increased to drive thedynamics of the hardware to a classical regime steady state ofdissipative dynamics that encodes the solution to the computationalproblem (step 404). After increasing the dechoerence, the dynamics ofthe dissipative-driven hardware can be simplified to:

$\frac{\rho}{t} = {{{- 2}H_{decoh}\rho} + {\sum\limits_{i,i^{\prime},j,j^{\prime}}\; {\Gamma_{i,i^{\prime},j,j^{\prime}}a_{i}^{\dagger}a_{i^{\prime}}\rho \; a_{j^{\prime}}^{\dagger}a_{j}}}}$

Local marginal probabilities can then be determined by the hardware andin some implementations a classical computer (step 406):

${\frac{{{tr}\left\lbrack {P_{m}\rho} \right\rbrack}}{t} = {{{- 2}\; {{tr}\left\lbrack {P_{m}H_{decoh}\rho} \right\rbrack}} + {\sum\limits_{i,i^{\prime},j,j^{\prime}}\; \Gamma_{i,i^{\prime},j,j^{\prime}}}}},{{tr}\left\lbrack {a_{j}^{\dagger}a_{j^{\prime}}P_{m}\; a_{j^{\prime}}^{\dagger}a_{i^{\prime}}\rho} \right\rbrack}$

where tr[ ] is the trace operation which in a density-matrix formulationis used to determine the expectation value of an operator, and P_(m) isa projector operator corresponding to the occupation density of a localcavity mode m.

The second term above retains density-density interactions betweenphotons in a cavity mode i and in a cavity mode j that contribute to thenumber of photons in the visible cavity mode m. The second term furtherretains the Γ_(i,i′,j,j′) tensor which can be related to a Markovtransition matrix, which is a matrix used in the problem if the problemcan be described as a Markov Random Field.

In some implementations, the problem, e.g., a probabilistic inference,can be encoded in a quantum probability distribution of the dissipativeBose-Hubbard Hamiltonian or its extended engineered version; that isusing the concept of quantum graphical models.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of anyinvention or of what may be claimed, but rather as descriptions offeatures that may be specific to particular embodiments of particularinventions. Certain features that are described in this specification inthe context of separate embodiments can also be implemented incombination in a single embodiment. Conversely, various features thatare described in the context of a single embodiment can also beimplemented in multiple embodiments separately or in any suitablesubcombination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various hardwaremodules and components in the embodiments described above should not beunderstood as requiring such separation in all embodiments, and itshould be understood that the described program components and hardwarescan generally be integrated together in a single software product orpackaged into multiple software products.

Particular embodiments of the subject matter have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results. As one example, the processesdepicted in the accompanying figures do not necessarily require theparticular order shown, or sequential order, to achieve desirableresults. In some cases, multitasking and parallel processing may beadvantageous.

What is claimed is:
 1. An apparatus comprising: a first plurality ofsuperconducting cavities each configured to receive a plurality ofphotons; a second plurality of superconducting cavities each configuredto receive a plurality of photons; and a plurality of couplers, whereineach coupler couples one superconducting cavity from the first pluralityof superconducting cavities with one superconducting cavity from thesecond plurality of superconducting cavities such that the photons inthe coupled superconducting cavities interact; and wherein a firstsuperconducting cavity of the first plurality of superconductingcavities is connected to a second superconducting cavity of the secondplurality of superconducting cavities, such that photons of the firstand second superconducting cavities are shared by each of the first andsecond superconducting cavities, the first superconducting cavity iscoupled to one or more of the other superconducting cavities of thefirst plurality of superconducting cavities to which the secondsuperconducting cavities are coupled, and the second superconductingcavity is coupled to one or more of the other superconducting cavitiesof the second plurality of superconducting cavities to which the firstsuperconducting cavities are coupled.
 2. The apparatus of claim 1,wherein each coupler is configured to annihilate a photon in onesuperconducting cavity and create a photon in a differentsuperconducting cavity.
 3. The apparatus of claim 1, wherein at leastone of the couplers comprises a Josephson Junction.
 4. The apparatus ofclaim 1, wherein a Hamiltonian characterizing the apparatus is:Σ_(i) h _(i) n _(i)+Σ_(i,j) t _(ij)(α_(i) ^(†)α_(j)+h.c.)+Σ_(i) U _(i) n_(i)(n _(i)−1), where n_(i) is a particle number operator and denotesoccupation number of a cavity mode i, α_(i) ^(†) is a creation operatorthat creates a photon in cavity mode i, α_(j) is an annihilationoperator that annihilates a photon in cavity mode j, h_(i) correspondsto a site disorder, U_(i) corresponds to an on-site interaction, t_(i,j)are the hopping matrix elements, and h.c. is hermitian conjugate.
 5. Theapparatus of claim 4, wherein the plurality of couplers is trained toproduce an output desired probability density function at a subsystem ofinterest at an equilibrium state of the apparatus.
 6. The apparatus ofclaim 4, wherein the apparatus is trained as a Quantum BoltzmannMachine.
 7. The apparatus of claim 1, wherein a Hamiltoniancharacterizing the apparatus is:${{\sum\limits_{i}\; {h_{i}n_{i}}} + {\sum\limits_{i,j}\; {t_{ij}\left( {{a_{i}^{\dagger}a_{j}} + {h.c.}} \right)}} + {\sum\limits_{i}\; {U_{i}{n_{i}\left( {n_{i} - 1} \right)}}} + {\sum\limits_{i,j}\; {U_{ij}n_{i}n_{j}}}},$wherein n_(i) is a particle number operator and denotes occupationnumber of a cavity mode i, α_(i) ^(†) is a creation operator thatcreates a photon in cavity mode i, α_(i) is an annihilation operatorthat annihilates a photon in cavity mode j, h_(i) corresponds to a sitedisorder, U_(i) corresponds to an on-site interaction, t_(i,j) are thehopping matrix elements, and h.c. is hermitian conjugate.
 8. Theapparatus of claim 7, wherein the apparatus is operable to evolveadiabatically to a ground state of a problem HamiltonianH_(p)=Σ_(i)h_(i)n_(i)+Σ_(i)U_(i)n_(i)(n_(i)−1)+Σ_(i,j)U_(ij)n_(i)n_(j).9. The apparatus of claim 7, wherein the apparatus is operable to evolveadiabatically from a Mott-insulator state to a superfluid state, andwherein an initial Hamiltonian of the apparatus isH_(i)=Σ_(i,j)t_(ij)(α_(i) ^(†)α_(j)+h.c.).
 10. The apparatus of claim 7,wherein the apparatus is operable to evolve adiabatically from aMott-insulator state to a ground state of a problem HamiltonianH_(p)=Σ_(i)h_(i)n_(i)+Σ_(i)U_(i)n_(i)(n_(i)−1)+Σ_(i,j)U_(ij)n_(i)n_(j),and wherein an initial Hamiltonian of the apparatus isH_(i)=Σ_(i,j)t_(ij); (α_(i) ^(†)α_(j)+h.c.
 11. The apparatus of claim 1,wherein the apparatus is configured to respond to an external field ε(t)and a Hamiltonian characterizing the apparatus in the external field is:Σ_(i) h _(i) n _(i)+Σ_(i,j) t _(ij)(α_(i) ^(†)α_(j)+h.c.)+Σ_(i) U _(i) n_(i)(n _(i)−1)+Σ_(i)[ε(t)α_(i) ^(†)+ε(t)*α_(i) ]+H _(SB),whereinH _(SB)=Σ_(i)Σ_(υ)[κ_(i,υ)(α_(i) b _(υ) ^(†)+α_(i) ^(†) b_(υ))+λ_(i,υ)α_(i) ^(†)α_(i)(b _(υ) +b _(υ) ^(†))], and wherein n_(i) isa particle number operator, ε(t) is a slowly-varying envelope of anexternally applied field to compensate for photon loss, H_(SB) is aHamiltonian of the interaction between the apparatus and a backgroundbath in which the apparatus is located, b_(υ), and b_(υ) ^(†) areannihilation and creation operators for a bosonic background bathenvironment, κ_(i,υ) is a strength of apparatus-bath interactionscorresponding to exchange of energy, h_(i) corresponds to a sitedisorder, U_(i) corresponds to an on-site interaction, t_(i,j) are thehopping matrix elements, and λ_(i,υ) corresponds to a strength of localphoton occupation fluctuations due to exchange of phase with the bath.12. The apparatus of claim 11, wherein the apparatus is operable to bedissipatively-driven to a ground state of a problem Hamiltonian.
 13. Theapparatus of claim 1, wherein at least one cavity is a 2D cavity. 14.The apparatus of claim 1, wherein each cavity is a 3D cavity.
 15. Theapparatus of claim 1, wherein each superconducting cavity in the firstplurality of superconducting cavities is connected to a superconductingcavity in the second plurality of superconducting cavities.
 16. A methodcomprising: providing the apparatus of claim 1 in an initialMott-insulated state; causing a quantum phase transition of theapparatus from the initial Mott-insulator state to a superfluid sate;and adiabatically guiding the apparatus to a problem Hamiltonian. 17.The method of claim 16, further comprising: causing a quantum phasetransition of the apparatus from the superfluid state to a finalMott-insulator state; and reading the state of each superconductingcavity in the apparatus.